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23 Mar 2013, 18:23

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Not necessary. If 2,3,5,7 are the factors of p, p is not a product of 4 (since 4 is not a factor of p). However, if 2,3,5, and 7 are the only prime factors of p, then p may be a multiple of 4 if the power of 2 in the product is 2 or more.
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25 Mar 2013, 09:03

GyanOne wrote:

Not necessary. If 2,3,5,7 are the factors of p, p is not a product of 4 (since 4 is not a factor of p). However, if 2,3,5, and 7 are the only prime factors of p, then p may be a multiple of 4 if the power of 2 in the product is 2 or more.

What if the question becomes, 2,3,5,7 are the only factor, not only prime factor. Is P then the product of the 4?

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Not necessary. If 2,3,5,7 are the factors of p, p is not a product of 4 (since 4 is not a factor of p). However, if 2,3,5, and 7 are the only prime factors of p, then p may be a multiple of 4 if the power of 2 in the product is 2 or more.

What if the question becomes, 2,3,5,7 are the only factor, not only prime factor. Is P then the product of the 4?

Hi score780, I'm afraid this question doesn't really make sense. 2, 3, 5 and 7 cannot be the only factors of some number p, because the number 1 is a factor of all numbers, and so is the number p itself. In this case, the number needs to be, at a minimum, 2x3x5x7 which is 210. 210 has a lot more factors than these 4 numbers. I'm pretty sure the question should mention that these are the only prime factors of p. If that's the case, then p can be the product of these four numbers, or any multiple of that number (i.e. 210, 420, 630, etc). So no if they are the only prime factors then it doesn't guarantee anything.

Be careful with factors, they always come in pairs (special case for perfect squares ) The factors of a number that has 2, 3, 5 and 7 as factors are at a minimum:

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Hi Ron, yes very clear. I am just confused about the pairs you mentioned at the end about perfect squares and pairs of factors. Can you please help me understand?

Hi score780, yes of course. I only wanted to draw attention to the fact that factors always come in pairs, so if you want to find all the factors of 20 you go find 1 and 20, 2 and 10 and 4 and 5. These numbers all multiply together to produce 20. The factors of 20 are thus, in order: 1, 2, 4, 5, 10 and 20. Each number has a pair with which it produces the original number.

For perfect squares (say 36), then the multiples become:1 and 362 and 183 and 124 and 96 and 6.

That is, all the factors are paired up, except for 6, which is paired up with itself. the factors of 36 are thus 1, 2, 3, 4, 6, 9, 12, 18 and 36. This is an odd number (nine factors) because it is a perfect square. 2 clearly pairs up with 18 to result in 36, and 9 pairs up with 4, but for perfect squares there will be one factor paired up with itself.